Complex line bundles with connection can be parametrized by unitary character of fundamental group Recently I'm reading the book Geometric Quantization and Quantum Mechanics and it states in chapter $3$ that

Assume $(X,\omega )$ is a symplectic manifold, $(L,\alpha )$ is a complex line bundle over $X$ with $\alpha $ the corresponding connection form on $L$ satisfying $d\alpha =-\pi^{\ast }\omega $, then the set of equivalence classes of such line bundles with connection can be parametrized by the unitary characters of the group $\pi_1 (X)$.

I just don't know why and where are references. Do anyone know about this fact?
 A: I think this can be directly proved.
First notice that the one-dimensional unitary representation of $\pi_{1}(X)$ can be noted by $\text{Hom}(\pi_{1}(X),S^{1})$. Notice that $\text{Hom}(\pi_{1}(X),S^{1})=\text{Hom}(H_{1}(X;\mathbb{Z}),S^{1})=H^{1}(X;S^{1})$ via abelianization and the fact that $\text{Ext}_{\mathbb{Z}}(H_{0}(X),S^{1})=0$.
On the other hand, the complex line bundle on the symplectic manifold $X$ is determined by $H^{2}(X;\mathbb{Z})$. And also we have the short exact sequence $0\rightarrow\mathbb{Z}\rightarrow\mathbb{R}\rightarrow S^{1}\rightarrow0$ and then we can obtain a long exact sequence $$\cdots\rightarrow H^{1}(X;\mathbb{Z})\rightarrow H^{1}(X;\mathbb{R})\rightarrow H^{1}(X;S^{1})\rightarrow^{f} H^{2}(X;\mathbb{Z})\rightarrow\cdots$$
Then we only need to show that for the equivalence of the line bundle $(L,\alpha)$ is uniquely determined by $H^{2}(X;\mathbb{Z})$. This is obvious since for $[\gamma]\in\text{Ker}(f)=\text{Im}(H^{1}(X;\mathbb{R})\rightarrow H^{1}(X;S^{1}))$, and the fact that $\alpha$ is equivalent to $\alpha+Z_{1}(L)$ and modulo the exact form we obtain that fix the line bundle $L$ the equivalence class of the connection form with the condition given above is determined by $H^{1}(L;\mathbb{R})$ and since the fiber of the vector bundle is contractible we have $H^{1}(L;\mathbb{R})=H^{1}(X;\mathbb{R})$. And then we can see that the equivalence class of the line bundle $(L,\alpha)$ can be parametrized by $H^{1}(X;S^{1})$, i.e. the unitary characters of $\pi_{1}(X)$.
